Journal of the Ramanujan Mathematical Society

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Some Explicit Computations in Arakelov Geometry of Abelian Varieties


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1 Universite Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
     

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Given a polarized complex abelian variety (A, L), a Gromov lemma makes a comparison between the sup and L2 norms of a global section of L. We give here an explicit bound which depends on the dimension, degree and injectivity diameter of (A, L). It rests on a more general estimate for the jet of a global section of L. As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and W¨ustholz on one hand and Bost on the other. They also improve some similar statements established by Graftieaux in 2000.
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  • Some Explicit Computations in Arakelov Geometry of Abelian Varieties

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Authors

Eric Gaudron
Universite Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Abstract


Given a polarized complex abelian variety (A, L), a Gromov lemma makes a comparison between the sup and L2 norms of a global section of L. We give here an explicit bound which depends on the dimension, degree and injectivity diameter of (A, L). It rests on a more general estimate for the jet of a global section of L. As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and W¨ustholz on one hand and Bost on the other. They also improve some similar statements established by Graftieaux in 2000.

References